A supplement to my paper on real and complex operator ideals

In this paper we study the maximal ideals in a commutative ring of bicomplex numbers and then we describe the maximal ideals in a bicomplex algebra. We found that the kernel of a nonzero multiplicative BC-linear functional in a commutative bicomplex Banach algebra need not be a maximal ideal. Finally, we introduce the notion of bicomplex division algebra and generalize the Gelfand-Mazur theorem for the bicomplex division Banach algebra.

This is the fourth of a series of papers surveying some small part of the remarkable work of our friend and colleague Nigel Kalton. We have written it as part of a tribute to his memory. It contains almost no new results. This time we discuss Nigel's partial solutions (obtained jointly with one of us) of the problem of whether the complex method of interpolation preserves the compactness of operators. This problem is now 51 years old and still lacks a complete solution. We al...

This paper surveys some of the late Nigel Kalton's contributions to Banach space theory. The paper is written for the Nigel Kalton Memorial Website http://mathematics.missouri.edu/kalton/, which is scheduled to go online in summer 2011.

In this article we give a short and informal overview of some aspects of the theory of C*- and von Neumann algebras. We also mention some classical results and applications of these families of operator algebras.

It is well known that the only proper non-trivial norm-closed ideal in the algebra L(X) for X=\ell_p (1 \le p < \infty) or X=c_0 is the ideal of compact operators. The next natural question is to describe all closed ideals of L(\ell_p\oplus\ell_q) for 1 \le p,q < \infty, p \neq q, or, equivalently, the closed ideals in L(\ell_p,\ell_q) for p < q. This paper shows that for 1 < p < 2 < q < \infty there are at least four distinct proper closed ideals in L(\ell_p,\ell_q), includi...

Let $X$ be a completely regular topological space. We assign to each (set theoretic) ideal of $X$ an (algebraic) ideal of $C_B(X)$, the normed algebra of continuous bounded complex valued mappings on $X$ equipped with the supremum norm. We then prove several representation theorems for the assigned ideals of $C_B(X)$. This is done by associating a certain subspace of the Stone--\v{C}ech compactification $\beta X$ of $X$ to each ideal of $X$. This subspace of $\beta X$ has a s...

For a certain class of algebras $\cal A$ we give a method for constructing Banach spaces $X$ such that every operator on $X$ is close to an operator in $\cal A$. This is used to produce spaces with a small amount of structure. We present several applications. Amongst them are constructions of a new prime Banach space, a space isomorphic to its subspaces of codimension two but not to its hyperplanes and a space isomorphic to its cube but not to its square.

We study in this paper the infinite-dimensional orthogonal Lie algebra $\mathcal{O}_C$ which consists of all bounded linear operators $T$ on a separable, infinite-dimensional, complex Hilbert space $\mathcal{H}$ satisfying $CTC=-T^*$, where $C$ is a conjugation on $\mathcal{H}$. By employing results from the theory of complex symmetric operators and skew-symmetric operators, we determine the Lie ideals of $\mathcal{O}_C$ and their dual spaces. We study derivations of $\mathca...

In this paper, we present an algebraic approach to idempotent functional analysis, which is an abstract version of idempotent analysis. The basic concepts and results are expressed in purely algebraic terms. We consider idempotent versions of certain basic results of linear functional analysis, including the theorem on the general form of a linear functional and the Hahn-Banach and Riesz-Fischer theorems.

Applying a linearization theorem due to J. Mujica, we study the ideals of bounded holomorphic mappings $\mathcal{H}^\infty\circ\mathcal{I}$ generated by composition with an operator ideal $\mathcal{I}$. The bounded-holomorphic dual ideal of $\mathcal{I}$ is introduced and its elements are characterized as those that admit a factorization through $\mathcal{I}^\mathrm{dual}$. For complex Banach spaces $E$ and $F$, we also analyze new ideals of bounded holomorphic mappings from ...